--- title: "Using a delay-adjusted case fatality ratio to estimate under-reporting" description: "Using a corrected case fatality ratio, we calculate estimates of the level of under-reporting for any country with greater than ten deaths" status: real-time-report rmarkdown_html_fragment: true update: 2020-08-18 authors: - id: tim_russell corresponding: true - id: joel_hellewell equal: 1 - id: sam_abbott equal: 1 - id: nick_golding - id: hamish_gibbs - id: chris_jarvis - id: kevin_vanzandvoort - id: ncov-group - id: stefan_flasche - id: roz_eggo - id: john_edmunds - id: adam_kucharski ---

Aim

To estimate the percentage of symptomatic COVID-19 cases reported in different countries using case fatality ratio estimates based on data from the ECDC, correcting for delays between confirmation-and-death.

Data availability

The under-reporting estimates for all countries can be downloaded as a single .csv file here.

Similarly, the prevalence estimates can be downloaded as a single .csv file here.

How to cite this work

If you wish to cite this work, please do cite the associated preprint [1]).

Methods Summary

The associated preprint[1], specifically the corresponding supplementary material contains a full description of the methods and limitations used to arrive at the estimates presented here.

Current estimates of under-reporting, prevalence and adjusted case curves along with reported cases

Temporal variation

Figure 1: Temporal variation in reporting rate. We calculate the percentage of symptomatic cases reported on each day a country has had more than ten deaths. We then fit a Gaussian Process (GP) to these data (see Temporal variation model fitting section for details), highlighting the temporal trend of each countries reporting rate. The red shaded region is the 95% CrI of fitted GP.

Prevalence estimates

Country Prevalence median (95% CrI) Total reported cases New reported cases (tallied over last 10 days) Population
Afghanistan 0.0095% (0.0045% - 0.023%) 36,326 1,067 38,928,341
Albania 0.19% (0.085% - 0.45%) 4,842 872 2,877,800
Algeria 0.034% (0.017% - 0.073%) 27,956 5,424 43,851,043
Angola 0.0069% (0.0031% - 0.017%) 836 245 32,866,268
Argentina 0.37% (0.2% - 0.7%) 162,482 43,225 45,195,777
Armenia 0.25% (0.13% - 0.5%) 37,339 2,752 2,963,234
Australia 0.063% (0.03% - 0.14%) 14,809 3,494 25,499,881
Austria 0.024% (0.014% - 0.05%) 20,448 1,042 9,006,400
Azerbaijan 0.073% (0.041% - 0.14%) 30,281 3,313 10,139,175
Bahamas 0.15% (0.078% - 0.38%) 334 244 393,248
Bahrain 0.41% (0.26% - 0.73%) 36,561 3,478 1,701,583
Bangladesh 0.03% (0.018% - 0.056%) 226,192 24,159 164,689,383
Belarus 0.061% (0.026% - 0.16%) 67,170 1,298 9,449,321
Belgium 0.056% (0.034% - 0.1%) 67,547 3,152 11,589,616
Benin 0.0037% (0.0017% - 0.0095%) 1,509 168 12,123,198
Bolivia 0.84% (0.45% - 1.6%) 71,792 13,673 11,673,029
Bosnia & Herzegovina 0.33% (0.16% - 0.73%) 10,468 2,334 3,280,815
Brazil 0.56% (0.31% - 1%) 2,442,362 367,515 212,559,409
Bulgaria 0.15% (0.071% - 0.35%) 10,529 1,983 6,948,445
Burkina Faso 0.00056% (0.00029% - 0.0019%) 1,080 48 20,903,278
Cameroon 0.0044% (0.0026% - 0.012%) 16,636 551 26,545,864
Canada 0.027% (0.016% - 0.053%) 114,552 4,939 37,742,157
Cape Verde 0.13% (0.071% - 0.34%) 1,631 314 555,988
Central African Republic 0.0051% (0.003% - 0.014%) 3,530 114 4,829,764
Chad 0.00057% (0.00025% - 0.0049%) 889 33 16,425,859
Chile 0.24% (0.12% - 0.88%) 347,722 19,077 19,116,209
China 0.00019% (0.00011% - 0.0012%) 86,724 1,300 1,439,323,774
Colombia 0.66% (0.37% - 1.2%) 257,056 66,401 50,882,884
Congo - Brazzaville 0.022% (0.013% - 0.046%) 3,057 567 5,518,092
Congo - Kinshasa 0.0017% (0.00073% - 0.0067%) 8,830 521 89,561,404
Costa Rica 0.28% (0.13% - 0.69%) 15,108 5,290 5,094,114
Côte d’Ivoire 0.013% (0.0083% - 0.024%) 15,332 1,743 26,378,275
Croatia 0.045% (0.02% - 0.13%) 4,675 646 4,105,268
Cuba 0.0017% (0.00096% - 0.0047%) 2,484 87 11,326,616
Cyprus 0.0044% (0.0024% - 0.011%) 914 23 1,207,361
Czechia 0.032% (0.019% - 0.06%) 15,302 1,661 10,708,982
Denmark 0.014% (0.0081% - 0.029%) 13,516 374 5,792,203
Djibouti 0.012% (0.0071% - 0.026%) 3,926 56 988,002
Dominican Republic 0.24% (0.15% - 0.49%) 64,151 12,637 10,847,904
Ecuador 0.17% (0.092% - 0.35%) 85,138 7,779 17,643,060
Egypt 0.032% (0.017% - 0.066%) 92,433 5,310 102,334,403
El Salvador 0.26% (0.13% - 0.58%) 14,871 3,527 6,486,201
Equatorial Guinea 0% (0% - 0%) 2,632 0 1,402,985
Estonia 0.0032% (0.0016% - 0.011%) 1,771 17 1,326,539
Eswatini 0.18% (0.083% - 0.45%) 1,826 587 1,160,164
Ethiopia 0.018% (0.0092% - 0.036%) 14,230 5,400 114,963,583
Finland 0.0035% (0.0018% - 0.013%) 7,131 80 5,540,718
France 0.027% (0.016% - 0.053%) 183,833 8,405 65,273,512
Gabon 0.08% (0.049% - 0.15%) 6,913 874 2,225,728
Georgia 0.0068% (0.0037% - 0.018%) 701 117 3,989,175
Germany 0.011% (0.007% - 0.019%) 206,113 4,668 83,783,945
Ghana 0.043% (0.026% - 0.075%) 33,246 6,564 31,072,945
Greece 0.0059% (0.0029% - 0.016%) 4,161 244 10,423,056
Guatemala 0.21% (0.11% - 0.42%) 44,966 7,011 17,915,567
Guinea 0.0088% (0.0054% - 0.017%) 6,367 564 13,132,792
Guinea-Bissau 0.0039% (0.0017% - 0.012%) 865 27 1,967,998
Guyana 0.033% (0.011% - 0.11%) 322 62 786,559
Haiti 0.024% (0.008% - 0.081%) 7,283 287 11,402,533
Honduras 0.27% (0.14% - 0.56%) 39,732 6,948 9,904,608
Hungary 0.0061% (0.0021% - 0.022%) 4,444 141 9,660,350
Iceland 0.011% (0.0059% - 0.028%) 292 16 341,250
India 0.059% (0.037% - 0.098%) 1,483,083 405,538 1,380,004,385
Indonesia 0.028% (0.015% - 0.054%) 100,301 15,421 273,523,621
Iran 0.27% (0.15% - 0.51%) 293,606 22,000 83,992,953
Iraq 0.22% (0.12% - 0.42%) 112,559 22,365 40,222,503
Ireland 0.015% (0.0059% - 0.041%) 25,822 142 4,937,796
Israel 0.35% (0.22% - 0.62%) 64,342 15,093 8,655,541
Italy 0.013% (0.0065% - 0.028%) 246,431 2,070 60,461,828
Jamaica 0.0071% (0.0033% - 0.029%) 319 79 2,961,161
Japan 0.0085% (0.0053% - 0.014%) 29,825 5,347 126,476,458
Jordan 0.0017% (0.00088% - 0.0048%) 278 72 10,203,140
Kazakhstan 0.16% (0.096% - 0.3%) 84,455 14,309 18,776,707
Kenya 0.024% (0.013% - 0.049%) 17,894 5,225 53,771,300
Kuwait 0.26% (0.16% - 0.46%) 63,636 5,475 4,270,563
Kyrgyzstan 0.24% (0.14% - 1.2%) 33,290 7,312 6,524,191
Latvia 0.0064% (0.0023% - 0.02%) 643 31 1,886,202
Lebanon 0.036% (0.02% - 0.082%) 3,783 1,107 6,825,442
Lesotho 0.037% (0.013% - 0.11%) 272 146 2,142,252
Liberia 0.0063% (0.002% - 0.042%) 1,119 79 5,057,677
Libya 0.069% (0.033% - 0.15%) 2,659 1,036 6,871,287
Lithuania 0.012% (0.0048% - 0.039%) 1,981 104 2,722,291
Luxembourg 0.3% (0.18% - 0.59%) 6,283 912 625,976
Madagascar 0.023% (0.013% - 0.047%) 9,078 2,841 27,691,019
Malawi 0.026% (0.012% - 0.06%) 3,271 802 19,129,955
Malaysia 0.00093% (0.00054% - 0.0026%) 8,805 140 32,365,998
Maldives 0.17% (0.1% - 0.33%) 1,232 439 540,542
Mali 0.00053% (0.00025% - 0.0019%) 2,495 41 20,250,834
Mauritania 0.032% (0.019% - 0.066%) 6,162 725 4,649,660
Mauritius 0% (0% - 0%) 8 0 1,271,767
Mexico 0.63% (0.35% - 1.2%) 395,463 56,576 128,932,753
Moldova 0.22% (0.11% - 0.45%) 23,074 2,360 4,033,963
Montenegro 0.57% (0.26% - 1.3%) 2,568 1,229 628,062
Morocco 0.045% (0.022% - 0.097%) 20,881 3,872 36,910,558
Mozambique 0.0019% (0.0011% - 0.0044%) 732 266 31,255,435
Namibia 0.058% (0.031% - 0.14%) 811 640 2,540,916
Nepal 0.0089% (0.0054% - 0.022%) 18,265 1,250 29,136,808
Netherlands 0.019% (0.011% - 0.035%) 53,144 1,570 17,134,873
New Zealand 0.00027% (1e-04% - 0.0011%) 435 4 4,822,233
Nicaragua 0.012% (0.0055% - 0.04%) 3,423 292 6,624,554
Niger 0.00034% (0.00014% - 0.0017%) 1,131 28 24,206,636
Nigeria 0.0053% (0.0031% - 0.01%) 41,083 5,073 206,139,587
North Macedonia 0.29% (0.14% - 0.63%) 10,172 1,205 2,083,380
Norway 0.0043% (0.0024% - 0.011%) 8,840 102 5,421,242
Oman 0.56% (0.28% - 2.3%) 76,574 11,554 5,106,622
Pakistan 0.011% (0.0066% - 0.021%) 275,195 11,729 220,892,331
Palestinian Territories 0.15% (0.095% - 0.27%) 12,756 3,870 5,101,416
Panama 0.72% (0.37% - 1.5%) 61,406 9,181 4,314,768
Paraguay 0.03% (0.016% - 0.08%) 4,374 919 7,132,530
Peru 0.26% (0.15% - 1.2%) 389,679 40,217 32,971,846
Philippines 0.031% (0.019% - 0.051%) 82,037 16,736 109,581,085
Poland 0.03% (0.015% - 0.065%) 43,380 3,656 37,846,605
Portugal 0.04% (0.023% - 0.13%) 50,439 1,909 10,196,707
Puerto Rico 0.31% (0.17% - 0.81%) 15,441 3,988 2,860,840
Qatar 0.28% (0.14% - 1%) 107,387 3,289 2,881,060
Romania 0.28% (0.15% - 0.56%) 45,877 9,211 19,237,682
Russia 0.11% (0.062% - 0.21%) 818,027 46,574 145,934,460
São Tomé & Príncipe 0.13% (0.07% - 0.34%) 653 122 219,161
Saudi Arabia 0.3% (0.15% - 0.61%) 268,801 20,518 34,813,867
Senegal 0.026% (0.012% - 0.06%) 9,387 1,095 16,743,930
Serbia 0.11% (0.055% - 0.27%) 24,095 3,643 8,737,370
Sierra Leone 0.0025% (0.0013% - 0.0087%) 1,733 82 7,976,985
Singapore 0.12% (0.068% - 0.25%) 49,959 3,183 5,850,343
Slovakia 0.0083% (0.0047% - 0.019%) 1,710 205 5,459,643
Slovenia 0.02% (0.0089% - 0.068%) 1,868 147 2,078,932
Somalia 0.00091% (0.00053% - 0.0024%) 3,170 67 15,893,219
South Africa 0.78% (0.43% - 1.5%) 452,255 101,650 59,308,690
South Korea 0.0019% (0.0011% - 0.0048%) 14,175 458 51,269,183
South Sudan 0.0027% (0.0013% - 0.0067%) 2,069 114 11,193,729
Spain 0.083% (0.05% - 0.27%) 279,864 18,527 46,754,783
Sri Lanka 0.00098% (0.00059% - 0.0019%) 1,936 101 21,413,250
Sudan 0.013% (0.0054% - 0.034%) 11,414 742 43,849,269
Suriname 0.19% (0.1% - 0.42%) 1,346 482 586,634
Sweden 0.048% (0.027% - 0.097%) 79,360 2,114 10,099,270
Switzerland 0.023% (0.014% - 0.044%) 34,372 984 8,654,618
Syria 0.0087% (0.003% - 0.024%) 487 178 17,500,657
Tajikistan 0.0087% (0.0053% - 0.02%) 7,235 401 9,537,642
Tanzania 0% (0% - 0%) 484 0 59,734,213
Thailand 0.00016% (8.6e-05% - 0.00044%) 3,120 48 69,799,978
Togo 0.0036% (0.0016% - 0.016%) 784 108 8,278,737
Tunisia 0.0021% (0.0011% - 0.0057%) 1,426 107 11,818,618
Turkey 0.023% (0.012% - 0.046%) 227,019 8,302 84,339,067
Ukraine 0.073% (0.037% - 0.15%) 65,651 7,545 43,733,759
United Arab Emirates 0.051% (0.031% - 0.11%) 59,024 2,466 9,890,400
United Kingdom 0.088% (0.047% - 0.17%) 300,592 5,413 67,886,004
United States 0.35% (0.22% - 0.59%) 4,290,248 578,799 331,002,647
Uruguay 0.018% (0.0069% - 0.051%) 795 158 3,473,727
Uzbekistan 0.031% (0.019% - 0.055%) 19,941 5,077 33,469,199
Venezuela 0.033% (0.02% - 0.06%) 15,687 4,505 28,435,943
Yemen 0.011% (0.0041% - 0.034%) 1,690 110 29,825,968
Zambia 0.11% (0.045% - 0.27%) 3,632 1,572 18,383,956
Zimbabwe 0.041% (0.02% - 0.09%) 2,153 1,226 14,862,927

Table 1: Estimates for the prevalence of COVID-19 in each country with greater than 10 deaths. We use the under-reporting estimates to adjust the reported case curves and tally these up over the last ten days as a proxy for prevalence. See Detailed Methods for more details.

Adjusted symptomatic case estimates

Figure 2: Estimated number of new symptomatic cases, calculated using our temporal under-reporting estimates. We adjust the reported case numbers each day - for each country with an under-reporting estimate - using our temporal under-reporting estimates to arrive at an estimate of the true number of symptomatic cases each day. The shaded blue region represents the 95% CrI, calcuated directly using the 95% CrI of the temporal under-reporting estimate.

Reported cases

Figure 3: Reported number of cases each day, pulled from the ECDC and plotted against time for comparison with our estimated true numbers of symptomatic cases each day, adjusted using our under-reporting estimates.

Current under-reporting estimates

Country Percentage of symptomatic cases reported (95% CI) Total cases Total deaths
Afghanistan 12% (8.1%-18%) 37,599 1,375
Albania 37% (26%-52%) 7,499 230
Algeria 76% (59%-92%) 39,025 1,379
Andorra 54% (21%-99%) 1,005 53
Angola 23% (16%-31%) 1,906 88
Argentina 46% (40%-52%) 294,556 5,750
Armenia 75% (61%-91%) 41,701 824
Australia 35% (27%-46%) 23,599 421
Austria 95% (75%-100%) 23,717 729
Azerbaijan 91% (78%-100%) 34,343 508
Bahamas 89% (62%-100%) 1,329 19
Bahrain 99% (93%-100%) 47,185 173
Bangladesh 91% (72%-100%) 279,144 3,694
Belarus 46% (29%-68%) 69,589 613
Belgium 97% (83%-100%) 78,441 9,944
Benin 78% (52%-100%) 2,063 39
Bolivia 43% (36%-50%) 101,223 4,123
Bosnia and Herzegovina 34% (25%-44%) 16,035 475
Brazil 58% (51%-64%) 3,359,570 108,536
Bulgaria 33% (24%-44%) 14,500 512
Burkina Faso 84% (42%-100%) 1,280 55
Cameroon 94% (57%-100%) 18,582 403
Canada 90% (70%-100%) 122,872 9,032
Cape Verde 89% (64%-100%) 3,203 36
Central African Republic 91% (47%-100%) 4,667 61
Chad 73% (16%-100%) 959 76
Chile 69% (39%-100%) 387,502 10,513
China 94% (26%-100%) 89,441 4,703
Colombia 41% (37%-46%) 476,660 15,372
Congo 94% (72%-100%) 3,835 76
Costa Rica 79% (57%-100%) 29,084 304
Cote dIvoire 99% (93%-100%) 17,026 110
Croatia 72% (44%-99%) 6,656 166
Cuba 91% (60%-100%) 3,364 88
Cyprus 87% (57%-100%) 1,351 20
Czechia 98% (89%-100%) 20,202 399
Democratic Republic of the Congo 20% (8%-52%) 9,705 242
Denmark 93% (74%-100%) 15,740 621
Djibouti 93% (74%-100%) 5,372 59
Dominican Republic 84% (58%-100%) 86,737 1,481
Ecuador 60% (49%-70%) 101,751 6,083
Egypt 26% (21%-31%) 96,590 5,173
El Salvador 58% (45%-72%) 23,193 618
Equatorial Guinea 86% (57%-100%) 4,821 83
Estonia 82% (42%-100%) 2,192 63
Eswatini 55% (39%-75%) 3,894 73
Ethiopia 51% (41%-61%) 31,336 544
Finland 79% (30%-100%) 7,752 334
France 96% (88%-100%) 219,029 30,429
Gabon 98% (91%-100%) 8,270 53
Gambia 21% (14%-30%) 1,872 63
Georgia 87% (58%-100%) 1,351 17
Germany 100% (96%-100%) 224,014 9,232
Ghana 99% (95%-100%) 42,653 239
Greece 75% (46%-100%) 7,222 230
Guatemala 47% (39%-57%) 62,944 2,389
Guinea 98% (88%-100%) 8,620 51
Guinea Bissau 72% (42%-99%) 2,117 33
Guyana 55% (29%-94%) 709 23
Haiti 16% (8.4%-28%) 7,897 196
Honduras 69% (56%-83%) 50,995 1,583
Hungary 36% (18%-65%) 4,970 609
Iceland 89% (57%-100%) 2,014 10
India 100% (100%-100%) 2,702,742 51,797
Indonesia 44% (37%-50%) 141,370 6,207
Iran 20% (18%-23%) 345,450 19,804
Iraq 59% (50%-67%) 180,133 5,954
Ireland 52% (32%-79%) 27,313 1,774
Israel 98% (88%-100%) 95,129 692
Italy 63% (49%-81%) 254,235 35,400
Jamaica 81% (43%-100%) 1,129 14
Japan 100% (98%-100%) 55,958 1,114
Jersey 37% (9.3%-95%) 357 32
Jordan 85% (51%-100%) 1,398 11
Kazakhstan 44% (34%-55%) 121,973 1,635
Kenya 82% (67%-96%) 30,365 482
Kosovo 25% (19%-32%) 11,373 391
Kuwait 99% (93%-100%) 76,827 502
Kyrgyzstan 95% (31%-100%) 42,146 1,498
Latvia 54% (29%-94%) 1,323 32
Lebanon 85% (60%-100%) 9,337 105
Lesotho 42% (24%-74%) 946 30
Liberia 27% (6%-80%) 1,277 82
Libya 45% (35%-58%) 8,579 157
Lithuania 71% (37%-100%) 2,436 81
Luxembourg 96% (81%-100%) 7,469 124
Madagascar 91% (75%-100%) 13,886 171
Malawi 38% (27%-59%) 5,072 161
Malaysia 94% (58%-100%) 9,212 125
Maldives 96% (85%-100%) 5,909 23
Mali 79% (38%-100%) 2,640 125
Mauritania 97% (78%-100%) 6,701 157
Mexico 12% (11%-14%) 525,733 57,023
Moldova 54% (44%-65%) 30,377 908
Montenegro 64% (45%-88%) 4,085 80
Morocco 48% (37%-60%) 43,558 658
Mozambique 90% (65%-100%) 2,914 19
Namibia 83% (54%-100%) 4,344 36
Nepal 72% (32%-100%) 27,241 107
Netherlands 99% (91%-100%) 63,424 6,163
New Zealand 61% (26%-99%) 1,293 22
Nicaragua 66% (27%-100%) 4,115 128
Niger 68% (23%-100%) 1,167 69
Nigeria 94% (82%-100%) 49,485 977
North Macedonia 48% (37%-64%) 12,774 545
Norway 90% (59%-100%) 10,004 261
Oman 72% (30%-100%) 83,226 588
Pakistan 94% (80%-100%) 289,832 6,190
Palestine 99% (94%-100%) 22,391 121
Panama 64% (53%-77%) 82,543 1,788
Paraguay 58% (33%-94%) 10,135 145
Peru 7.1% (4.4%-11%) 541,493 26,481
Philippines 100% (100%-100%) 164,474 2,681
Poland 71% (57%-87%) 57,279 1,885
Portugal 93% (49%-100%) 54,234 1,779
Puerto Rico 91% (62%-100%) 26,760 335
Qatar 71% (30%-100%) 115,368 193
Romania 35% (29%-41%) 71,194 3,029
Russia 55% (48%-62%) 927,745 15,740
San Marino 74% (12%-100%) 719 42
Sao Tome and Principe 88% (55%-100%) 885 15
Saudi Arabia 37% (30%-48%) 299,914 3,436
Senegal 52% (37%-70%) 12,237 256
Serbia 81% (58%-98%) 29,782 677
Sierra Leone 82% (36%-100%) 1,956 69
Singapore 94% (72%-100%) 55,838 27
Sint Maarten 42% (14%-97%) 326 17
Slovakia 92% (70%-100%) 2,907 31
Slovenia 51% (20%-94%) 2,438 124
Somalia 92% (55%-100%) 3,257 93
South Africa 49% (41%-62%) 589,886 11,982
South Korea 95% (70%-100%) 15,761 306
South Sudan 78% (52%-99%) 2,490 47
Sri Lanka 96% (80%-100%) 2,900 11
Sudan 20% (13%-30%) 12,410 803
Suriname 79% (52%-100%) 3,077 48
Sweden 91% (78%-99%) 85,045 5,787
Switzerland 98% (88%-100%) 38,156 1,715
Syria 37% (20%-67%) 1,764 68
Tajikistan 97% (69%-100%) 8,099 64
Thailand 84% (52%-100%) 3,378 58
Togo 60% (19%-99%) 1,154 27
Tunisia 85% (46%-100%) 2,185 56
Turkey 87% (72%-100%) 250,542 5,996
Ukraine 58% (48%-69%) 94,436 2,116
United Arab Emirates 97% (78%-100%) 64,541 364
United Kingdom 22% (19%-25%) 319,197 41,369
United States of America 73% (49%-100%) 5,438,325 170,497
Uruguay 55% (32%-89%) 1,457 40
Uzbekistan 99% (93%-100%) 36,100 240
Venezuela 97% (88%-100%) 34,802 288
Yemen 6.4% (3.8%-10%) 1,882 535
Zambia 68% (47%-88%) 9,839 264
Zimbabwe 36% (27%-46%) 5,308 135

Table 2: Estimates for the proportion of symptomatic cases reported in different countries using cCFR estimates based on case and death timeseries data from the ECDC. Total cases and deaths in each country is also shown. Confidence intervals calculated using an exact binomial test with 95% significance.

Adjusting for outcome delay in CFR estimates

During an outbreak, the naive CFR (nCFR), i.e. the ratio of reported deaths date to reported cases to date, will underestimate the true CFR because the outcome (recovery or death) is not known for all cases [6]. We can therefore estimate the true denominator for the CFR (i.e. the number of cases with known outcomes) by accounting for the delay from confirmation-to-death [2].

We assumed the delay from confirmation-to-death followed the same distribution as estimated hospitalisation-to-death, based on data from the COVID-19 outbreak in Wuhan, China, between the 17th December 2019 and the 22th January 2020, accounting right-censoring in the data as a result of as-yet-unknown disease outcomes (Figure 1, panels A and B in [8]). The distribution used is a Lognormal fit, has a mean delay of 13 days and a standard deviation of 12.7 days [8].

To correct the CFR, we use the case and death incidence data to estimate the proportion of cases with known outcomes [2,7]:

\[ u_{t} = \frac{ \sum_{j = 0}^{t} c_{t-j} f_j}{c_t}, \]

where \(u_t\) represents the underestimation of the proportion of cases with known outcomes [2,6,7] and is used to scale the value of the cumulative number of cases in the denominator in the calculation of the cCFR, \(c_{t}\) is the daily case incidence at time, \(t\) and \(f_t\) is the proportion of cases with delay of \(t\) between confirmation and death.

Approximating the proportion of symptomatic cases reported

At this stage, raw estimates of the CFR of COVID-19 correcting for delay to outcome, but not under-reporting, have been calculated. These estimates range between 1% and 1.5% [2–4]. We assume a CFR of 1.4% (95% CrI: 1.2-1.7%), taken from a recent large study [4], as a baseline CFR. We use it to approximate the potential level of under-reporting in each country. Specifically, we perform the calculation \(\frac{1.4\%}{\text{cCFR}}\) of each country to estimate an approximate fraction of cases reported.

Temporal variation model fitting

We estimate the level of under-reporting on every day for each country that has had more than ten deaths. We then fit a Gaussian Process (GP) model using the library greta and greta.gp. The parameters we fit and their priors are the following: \[ \begin{aligned} &\sigma \sim \text{Log Normal(-1, 1)}: \quad &\text{Variance of the reporting kernel} \\ &\text{L} \sim \text{Log Normal(4, 0.5)}: \quad &\text{Lengthscale of the reporting kernel} \\ &\sigma_{\text{obs}} \sim \text{Truncated Normal(0, 0.5)}, \quad &\text{Variance of the obseration kernel, truncated at 0} \end{aligned} \] The kernel is split into two components: the reporting kernel \(R\), and the observation kernel \(O\). The reporting component has a standard squared-exponential form. For the observation component, we use an i.i.d. noise kernel to acccount for observation overdispersion, which can smooth out overly clumped death time-series. This is important as some countries have been known to report an unusually large number of deaths on a single day, due to past under-reporting.

In the sampling and fitting process, we calculate the expected number of deaths at each time-point, given the baseline CFR. We then use a Poisson likelihood, where the expected number of deaths is the rate of the Poisson likelihood, given the observed number of deaths

Approximating prevalence

We use the adjusted case curves, adjusted for under-reporting and for asymptomatic infections as a proxy for prevalence. Specifically, we tally up the adjusted new cases each day over the last ten days and calculate what percentage of the population in question this total equates to. This serves as a crude prevalence estimate. We assume ten days of infectiousness as taken from the mean of the total infectious period [9].

Adjusting case counts for under-reporting

We adjust the reported number of cases each day, pulled from the ECDC. Specifically, we divide the case numbers of each day by our “proportion of cases reported” estimates that we calculate each day for each country.*

Limitations

Implicit in assuming that the under-reporting is \(\frac{1.4\%}{\text{cCFR}}\) for a given country is that the deviation away from the assumed 1.4% CFR is entirely down to under-reporting. In reality, burden on healthcare system is a likely contributing factor to higher than 1.4% CFR estimates, along with many other country specific factors.

The following is a list of the other prominent assumptions made in our analysis:

Code and data availability

The code is publically available at https://github.com/thimotei/CFR_calculation. The data required for this analysis is a time-series for both cases and deaths, along with the corresponding delay distribution. We scrape this data from ECDC, using the NCoVUtils package [10].

The under-reporting estimates for all countries can be downloaded as a single .csv file here.

Similarly, global prevalence estimates can be downloaded as a single .csv file here

Acknowledgements

The authors, on behalf of the Centre for the Mathematical Modelling of Infectious Diseases (CMMID) COVID-19 working group, wish to thank DSTL for providing the High Performance Computing facilities and associated expertise that has enabled these models to be prepared, run and processed and in an appropriately-rapid and highly efficient manner.

References

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2 Russell TW, Hellewell J, Jarvis CI et al. Estimating the infection and case fatality ratio for covid-19 using age-adjusted data from the outbreak on the diamond princess cruise ship. medRxiv 2020.

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10 Abbott S MJ Hellewell J. NCoVUtils: Utility functions for the 2019-ncov outbreak. doi:105281/zenodo3635417 2020.